Why is the fraction of two logarithms for any base the same

Question

$$\frac{\log_{10} 100}{\log_{10} 1000} = \frac{\ln 100}{\ln 1000} = \frac{\log_{2} 100}{\log_{2} 1000}$$

Just curious why this is the case and wondering if there is a proof for it. Thanks

Answer 1

Simply because $$\log_b a = \frac{\log_c a}{\log_c b}$$ for any positive $a,b,c$ with $b\neq 1$, $c\neq 1$.

To see this, note

$$b^{\log_b a} = a\tag{defn of $\log_b$}$$ $$\log_c(b^{\log_b a}) = \log_c a\tag{apply $\log_c$ to both sides}$$ $$\log_b a\cdot \log_c b = \log_c a\tag{exponent rule for logs}$$ $$\log_b a = \frac{\log_c a}{\log_c b}\tag{div both sides by $\log_c b$}$$